Schedule:

We will meet until August 29, 2019, inclusive.

Huron Classroom Thursdays 4:00 – 6.00

About:

  1. I will try to discuss a backbone of the topics presented in the first five chapters of the book. Sometimes I will skip whole sections completely for the sake of time so that we can get an idea of what the topic is about. This is not so terrible since many results are familiar from other areas (i.e. several construction in Lie algebras) and we have many analogies with the known case of smooth manifolds.
  2. I try to update this page frequently and upload the notes I use whenever I use them. Notice that my notes and my presentation may differ in order or content slightly, but if you miss one lecture and see what happens in the notes you will be fine next time regardless.
  3. I am not an expert in this topic, I am learning as well so please tell me mistakes or comments or references I could add.
  4. We begin at 4.10 but from 4:00 to 4.10 I talk about a “nice fact” of something wlse that I am reading, not necessarily of mathematics.

Bibliography:

[1] p -adic Lie Groups, Peter Schneider.

[2] Arithmetic Differential Operators over the p-adic Integers, Claire C. Ralph, Santiago R. Simanca

Mostly for comparison and review of topology:

[3] Introduction to Topological Manifolds, Second Edition, John M. Lee, Springer GTM 202

[4] Introduction to Smooth Manifolds, First Edition, John M. Lee, Springer GTM 218

Logbook:

DateWhat did we do?Notes
July 4 We discussed mostly the concept of spherically
complete spaces and lemma 1.4
[1]: Section 1.1.
N/A
July 11 We talked about Mahler Expansion Theorem.
[2]: Chapter 3, pg 23 to 29.
PDF
July 18 We discussed in a different order most of the
material about power series
[1] Section 1.5
[1] Section 1.6, discussed only the beginning.
PDF
July 25 We began manifold theory and proved halfway
of Theorem 8.7 of [1] (i.e. up to page 16 of the
notes).
[1] Section 2.7, 2.8
PDF
August 1We saw up to page 12 of the notes. We defined
tangent plane, tangent bundle and Lie Group.
[1] Section 2.9, only the constructions of tangent
plane and tangent bundle.
[1] Section 3.13, results 13.1 to 13.5
PDF
August 8We saw the notes in its entirety: we studied the
algebraic properties of p-valuations.
[1] Section 5.23, 5.25.
PDF
August 15I improvised the lecture around the topics of section 26
of [1]. Of particular importance are the definition of
saturated, Lemma 26.13, Proposition 26.15 and 26.16, all
of [1].
N/A
August 22 We started the proof of theorem 27.1 of [1]. We
constructed the open compact subgroup G’.
PDF
August 29We finished the proof of theorem 27.1 of [1]. We constructed the valuation and proved its
properties.
PDF

Notes:

  • [1] Section 1.2: In here the continuos dual space is defined. We skipped this section since we use our intuition of the archimedean case and change it as we need (we do this just for the sake of time!).
  • [1] Section 1.3: Deals with convergence. In particular it proves the concept of conditional convergence doesn’t exist in this case.
  • Inverse Function Theorem: Compare the proof of the inverse function theorem ([1], Proposition 4.3) based on [1] Lemma 4.2, with the classical case in [4], Theorem 7.6.
  • Point of expansion: We did not prove Point of expansion ([1] Corollary 5.5) but the proof is instructive.
  • Characteristic: So far, we have made no distinction by field characteristic but some theorems and results require it or have a different proof. See for example [1] Corollary 5.7, 5.8.
  • Paracompactness: To review this concept see [3], chapter 4. For us [3] Lemma 4.80 and [3] Theorem 4.81 are the important ones. Notice that part of the proof of [1] Theorem 8.7 is exactly [3] Lemma 4.84. What makes the difference then in the proof is the third claim in (i) implies (ii).

Alternative References:

#Reference:
1Summary on non-Archimedean Valued Fields
Angel Barría Comicheo, Khodr Shamseddine
Advances in Ultrametric Analysis, AMS, CONM, Vol 74
2A journey through the history of p-adic numbers
Yvette Perrin
Advances in Ultrametric Analysis, AMS, CONM, Vol 74
3 Lectures on Lie groups over local fields
Helge Glockner
4On p-saturable groups
Jon González-Sánchez
5Continuous representation theory of p-adic Lie Groups
Peter Schneider